THE SEMILATTICE WITH CONSISTENT RETURN

8/12/2019 THE SEMILATTICE WITH CONSISTENT RETURN

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T H E SE lV rrL A T T I C E \-V IT H C O N S I S T E N T R E T U R N by I o n m U a c e no iu , D epa rtm e n t o f M ath em atics, U niversity o fC ra io v a ,

1100, R om ania

L e t p be a p rim e num ber. In [ ] is defined th e fu nc tion Sp as Sp :N * -- --+ N * ,S p(a) = k, w h ere k is the sm a lle s t p os itiv e in te g e r so th a t p a is a d iv iz o r for k L

A S m a ra n d a c h e func tion o f first k ind is d efined for each 11 E N* in [1], as nurnericaI func tion n :N ---- N s o that:

i) i f n = u i , w h ere u = 1 o r u = p , th en S1/ a ) = k, k b ein g the sm a lles t po sitive in teg e r w ith the p ro pe rty th a t k = M . u i a .

11 ) l f n pil pi2 .. , _pir then = I . 2 . r '

I t is p ro v e d that:

S n a + b ) ; S1 a ) . S n ( b )

I n [2] is p ro v e d th at:

i) th e fu nc tion Sn is m ono tonously increasing , ii) th e se q u e n c e o f func tions { } is m o no to n o us ly increasing .

p ie N

iii) fo r p , q - p rim e nurnbers such that: p < q S p < Sq a n d p i < q S < S q . w h ere i E N iv ) i f n < p , th en Sn < S p -I n [3] it is pro ved :

i) f o r p ~ 5 , Sp > m a x{Sp _ l, Sp + l}

ii) fo r p , q - p rim e num bers, i j E N

p < q and i:::; j S i < S J P q

iii) the s e q u e n c e o f functions {S n t e N is ge n e ra ly increasin g bo u nd le d iv ) i f n = p :l . p ; . / ; there are k k 2 , ... , km E {1 2 .. . , r} so that for ~ c h t E I , m there IS

E N so th a t


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nd for each l N we have:

SnU) = max{spi . I)}.-J . 5 t ~ J I l k k

(

We define the set { p ~t E 1, }as the set of active factors of n- and the others factors as the pasivefactors.

et NpI.P2-.-Pr ={n = p{l.p ~ 2 - P : l i l i 2... ,ir EN*}, where PI < P2 < .. < Pr are prime numbers.

Then

NPIP2-P { I h il ~ . ir t J: : }. r = 11 E PJP2 Pr n as PI ,P i , ,P r as ac I V ~lactorsis the S-active cone.

A Smarandache function of second kind is defined for each k EN* in [1] as the functionSk:N* 7 N* where Sk (n) =Sn k).

It is proved that:

Sk(a b) Sk(a). Sk(b)

n [ ] it is proved that:

i) for k,1l EN* the formula S k n ) ~ n . kis true

ii) all prime numbers P 5 are maximal points for Sk and

iii) the function Sk has its relative minimum values for every n =p , where P is a prime numberand P max{3,k}

iv) the numbers kp for p prime number, k EN* and p > k, are the fixed points of Sk

v) the function Sk have the following properties:

a) Sk= 0 nIH), for > 0

Sk(n)b) lim sup = k

n-+oo nc) Sk is, generally speaking , incresing, thus:

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1 DEFINITION Let g-ft = {Sm n)\n,m EN*}, *et A,BEP? N \ 0 and a = min A,

b = min B, a *= max A, b*=max B . The set J is the set o f he functions:

Sa h),n < max{a,h}

Sak bk , max {a, b} ~ n::; max{ak ,b k }

whereB * . B }JA:N ~ g - f t wzth JA n)= a k m ~ai E A l a i ~ J 1

I

c = ~ {bj EBlbj }J

* * *a. b ),n> max{a ,b }

2. EXAMPLES

) 6 O,12} N g-f t da 38IO}: an :\ , ,

n 1 2 3 4 5 6 7 8 9 10 11 12 n>13I{6,IO,12}

{3,8,lO} S3 6) S3 6) S3 6) S3 6) S3 6) S3 6) 3 3 6) S8 6) 3 8 6) 8 10 10) 310 10) SIO 12) SIO 12)

b) Let A = {1,3,5, ... , 2k+l , .. }

B = {2,4,6, ... ,2k, .. }

n 1 2 3 4 5 6 2k 2k+lIBA S2 l) S2 1) S2 3) S4 3) S4 5) S6 5)

c) Let A = {5,9,lO} and I : 1 , I i ~ : : Ng-ft with

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nA

IAN

IN

2 3 4 5 6 7 8 9

S5 5) S5 5) S5(5) S5(5) S5(5) S5(5) S5(5) S5(5) S9 9)

SI (1) S2 2) S3(3) S4(4) S5(5) S6(6) S7(7) S8(8) S9 9)

t is easy to see that 11 is not the reduction of IN: and 11cN*) c IN: (N*).N . N

3. REMARK.

The functions whi tch belongs to the set have the folowing propert ies:

1 i f Al c A2 and n E A l , then ~ (n) = 1;;2 (n)

1 ) i f ~ c ~ and n ~ , then I . ~(n) = .1fz (n)

10 n> 11

SlO 10) SlO lO)

S1O 10) SnCn)

2) IN: 11) =Sn n) =Sn n), the function IN: is called the I - diagonal function and IN: CN* isN N Ncalled the diagonal o f g f{ .

3) for each m EN* I{N} = Sm for I{N}(n) = Sm(n), \In EN*.. m m

3 ) for each m EN* ~ : }= Sill for I ~ : } n )= Sn(m) = Sm(n), \In EN*,

4) i f n EA lB , then IiCn) = ~ ? C n=SnCn).4. DEFINITION. For each pair m,n EN*, SmCn) and SmCn) are called the simetrical numbers

relative to the diagonal o f

Sm and Sill are called the simmetrical functio:n.s relative to the I-diagonal function I ~ : .As a rule,

I and I ~ are called the sirnrnetrical functions relative to the I-diagonal [unction I ~ : .5. DEFINITION. Let us consider the following rule T:I x I I, I . ~T I@ = . ~ ~ g .It is easy to see

that T is idempotent, commutative and associative, so that:

i) I i T 15f = I i

ii) I;; T I@ = I@ T 15f

iii) (I;; T I@) T f = 15fT I@T I f , where A,B,C,D,E,F E?PCN*)\0

6. DEFINITION. Let us consider the following relative partial order relation p, where:

p c l x I ,

~ p I g A c ( ' and B c D.

It is easy to see that CI, T,p) is a semilattice.

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7. DEFINITION. The elements l l , v E I are p - preceded i f here is w E I so that:

W pu and w pv.

8. DEFINITION. The elements u, v EI, are p - strictly preceded by w if:

i) W pll and w pv.

ii) \Ix EI\{w} so that x pll and x p v =>x pw.9. DEFINITION. Let us defined:

l ={(u, v) EI x Ilu, v are p preceded}1 ={(ll, v) E I x u v are p - strictly preceded}.

t is evidently that (u, v E 1 ~ (v, u) E l a n d (ll, v) E 1 (v,u) E 1 .10. DEFINITION. Let us consider T =U xU, U c I and let us consider the following rule:

1.:1 4 W, W e 1, i 1. Ig = i ~ g and the ordering partial relation r c U x U so thati r I g ~ I gp i

The structure (1 ,.l,r) is called the return of semi lattice 1, T,p).11. DEFINITION. The following set

is called the base o f return (1 ,.l,r).

12. REMARK. The base of return has the following properties:

i) if IB Ere => 1. 1 EreA Bii) for 0:t; X c N*,I Ere

iii) for i Ere is true the following equivalence 0:t; X c C N . A I \ B ) ~nonexistence of . l I . ~ .

B * B 13. PROPOSITION. For IA ~ there e.xists n E N so that Irl, (n) = l ~ .(n).

Proof Because A l B :t; 0 it results that there exists n E A 1\ B so that:

It results that for I ~ ~ then I ~has at least a point of contact with I-diagonal function.

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and

14 REMARK From the 1 it results:

b, n b5:n5:b =maxB

where

bk = max{x E \X5: n},n>b

a, ma

15. PROPOSITION There are troe thefollowing equivalences:

liCn) = I t} Cn) =SnCbn ), IS Cn) = { ~ } C n )=SnCdn ), liCm) = ~ I 1 } C m )=S llCam), andIgCm) = I ~ : } C m )= S"\c m) where am' b m c,mdn are defined in the sense o f 14.

I f n ~ m ,then n ~ a mC m ~ m .

Proof Evidently,

I , Ig) E 1# n C t 0 and B n D t 0 l ~ ,I f E P J

Because n C t 0 and B n D t 0 it exists n E n C and m E B n D Then:

IiCn) = { ~ } C n )=SnCbn ), Ig Cn) = { ~ } C n )=SnCdn

)

IiCm) = I);n}(m) =SmCam), Ig Cm) = bm}Cm)=S '(c m).Conversely, i f there exist n E N so that IiCn) =SnCbn ) and Ig Cll) =SnCdn ), then because

liCn) =SnCbn ) it results n =ak = ~ { iEAlai ~ n}, so that 11 EA. Because IgCI1) =SnCdn ) it resultsI

nEC.

Therefore AnC:t: 0 , thus, finally, I J Ere. t is also proved If PlJ in the some way.I f n 5: m, because n E n C it results that n E {x E Alx 5: m} and n E {y E qy 5: m}, therefore

n 5: am 5: m and ~ cm 5: m.

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This is presented in the following scheme:

n m

I ~

m

S,- (n) .

16. DEFINITION The retum (L ,J..,r) o/semillatice (L,T,p)Js:a)nul l i f L ={(u,u)luEL}=LlL.

b) weak, i f card[,# < card(L x L \ [,#)c) consistent, if cardL# = card(L x L - L )

d) vigour, i f cardL# > card(L x L - L )

e) total, i f L = L x L.

17. PROPOSITION The retum (r,J. . ,r) o/the semilattice (I, T,p) is consistent.

Proof Evidently, card(g>(N*)\ ) = ~ ,card I = card[(g>(N*)-0) x C??J>(N*)- 0)] ~ and

card(l x I) = K

Let us consider a--={ A,C)IA,Ceg.> N*)-0,AnC=0} and Gff={ A,C)IA,CEg> N*)-0,

AnC:;t;0}.

cardgt; c a r d ~= ~ Indeed, i f A n C = it results that CNoAuCNoC=N ; because for every

X EP N*) - 0 = Y = N* \ X so that X u Y = N* then it results c a r d ~ - ; -= card ~ N * )= ~ .Because for

each A,C), A,C EgJJ(N*)- 0 An C = 0 it exist at kast two AJ,C l ), A2 ,C 2 ) with

Ai n C l :;t; 0 A2 n C2 :;t; 0 it results c a r d 5 c a r d ~= ~

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THE SEMILATTICE WITH CONSISTENT RETURN

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